Recall that, for the \(q_h\) -subproblem (REF ), we use the warm start technique for the PCG executions, and the implementation of MG V-circles is based on the iFEM package developed in [1]} with Jacobi splitting.
Moreover, for the \(p_h\) -subproblem (), we use the pre-trained deep CNNs in [2]} (https://github.com/cszn/DnCNN) and the mapping \(\mathcal {M}\)  in Subroutine  is specified as \(\mathcal {M}(\mathbf {p})=(\mathbf {p}-\mathbf {a}_0)/(a_1-a_0)\times 256\) , where \(\mathbf {p}\)  is the coefficient of \(p_h\)  and \(I\)  is a raster image. The initial guess of the Lagrange multiplier \(\mathbf {\lambda }^0\)  is always set to be \(\mathbf {0}\) ; the initial guess of \(\mathbf {q}^0\) , \(\mathbf {u}^0\)  and \(\mathbf {p}^0\)  are set to be \(\mathbf {1}\) , \(\mathbf {1}\)  and \(\mathbf {0}\) , respectively. For the stopping criterion to solve the linear systems (REF ) and (REF ), relative errors are controlled with the tolerances of \(10^{-10}\)  and \(10^{-5}\) , respectively.
In addition, the value of \(\text{``Tol''}\)  in Subroutine REF  is \(10^{-3}\) .
